# Purpose of using polyphase filter bank (PFB)

What is the advantage of using a polyphase filter bank (PFB) for spectral analysis over just using the FFT? In the standard "critically sampled" uniform DFT filterbank, the polyphase decimation/filtering is followed by an $M$-point DFT block, which implements the last step of the PFB in a computationally efficient manner using the FFT.

If you have to do an FFT anyways, why bother with the PFB? Is the reason that I can choose a custom prototype low-pass filter on the front-end? Are there some computational savings I'm missing out on?

EDIT: If comparing this to a bank of quadrature downconverters, what is the point of using a PFB if the FFT is the same mathematically? It can't be the delay of $N$ samples requried to fill up an FFT block because the decimated branches have $1/N$ the rate, which means you will be waiting the same amount of time on average with either approach. What am I missing?

## 2 Answers

One advantage of a polyphase filterbank approach is, as you guessed, that you can control the frequency response of each channel. When using a DFT alone, you have limited control over the frequency band covered by each bin (characterized by a Dirichlet kernel in the unwindowed case, or by the frequency response of whichever window function you select). This is sufficient for a lot of applications.

In some other applications, however, you might want tighter control over the per-channel frequency response. Say you wanted to construct a DFT-based spectrum analyzer with very particular specifications (e.g. -3 dB response at the midpoint between output bins, -80 dB response at a one-bin spacing from center). You can utilize the polyphase filterbank structure to implement whatever filter is needed to achieve the specified level of performance.

Another application is in cases where reconstruction is needed after the channelizer: if you're careful in the design of the polyphase filter, you can actually perform straightforward spectral modification (similar to the "ideal filtering" that many signal processing beginners attempt using the DFT) in the frequency domain, then use a synthesis filterbank to take the composite signal back into the time domain. This structure, with cascaded analysis and synthesis stages, is known as a transmultiplexer.

The PFB is not really for spectral analysis. The primary feature of the PFB is that you can use it to create multiple independent streams from the FFT output as if you had multiple independent quadrature downconverters, while being much more computationally efficient than the quadrature downconverter approach.

• PFBs are a great approach to spectral analysis - FFTs have terrible spectral response (frequency leakage between channels). – Chris Jun 4 '19 at 5:23
• @Chris You’re right. I was surprised to read my own answer. I guess I must have learned that in between then and now. I’ll change my answer when I have time. – Jim Clay Aug 30 '19 at 15:55